The shift from a static knowledge to a dynamic one.

1.  Mathematics activities
 Science activities
Comenius 2003/2006 Nobel project
Comenius 2007/2009 SMILE project
Scuola Media Luigi Stefanini
Teacher Graziano Scotto di Clemente GP Moroldo F. Tronchin

2. Geometrical activities
The geometrical image of the world
 From drawing in pencil to 3D CAD
 A research on polyhedra (from Platon
to Euler) and the Internet resources
 An hypertext

3. The strands
 Give a variety of challenges and tools (physical objects,
pen, ruler and compass, DGS), to take account of the
pupils’ different attitudes and learning styles(
challenge, technology, support)
 Stimulate interest, curiosity, motivation ( fun,
motivation)
 apply mathematical knowledge to real world situations
( real world)
 Invite children to develop their own projects and to share
each other ideas and knowledge (self guided learning,
cooperation)

4. We have worked on:
 Drawings in axonometric projection many common
solids: prisms, pyramids, cones, cylinders…
 Designing with a simple 3D CAD the same solids and
some objects
 Exploring the Internet resources to search interactive
applications and informations about some particular
polyhedra
 Studying the geometrical properties of these polyhedra
until to find the Euler’s rule
 Solving traditional problems on volumes, area….

5. Solid geometry and “traditional
aspects” in the Italian curriculum
 Volume, surfaces, area and other
geometrical properties (apothem…..and so
on..)
The three-dimensional
representation:
the axonometric
projection

6. Solid geometry and ICT: We are testing a
simple 3D CAD to design polyedra and virtual objects.
The pupils have:
1. To fix the shape of solid base
2. To extrude the height
3. To define the kind of material
4. To choose the plane (x,y,z) of representation
5. ….

7. Solid geometry and ICT
Some resources from Internet to display
better the third dimension
• WRL animations
• CABRI animations
• Java animations
• Two simple free software:
 Cartesio (CAD from Venice University)
 and Poly

8. Some stages in the history of
geometry
 The Platonic solids: cube, tetrahedron,
octahedron, dodecahedron and
icosahedron
 The 13 Archimedean solids
 The Euler’s rule
(f = faces number; v= vertices number; s = edges
number)

9. To stimulate interest, curiosity,
motivation: some links between
geometry and art
 Art and geometry during the Italian
Renaissance
 Escher
 Some images in the next slide
All these things are present
in our hypertext, synthetically.

10. Leonardo da Vinci Giulia
Escher

11. Dynamic geometry
 Search geometry in the real world
(for pupils 11-12 years old)
Using LOGO Using Cabri II Plus

12. Dynamic geometry and LOGO
Free Software:
Logo for Microsoft Windows,
Softronics Inc,
www.softronix.com
The Turtle’s motion and the construction of shapes:
Some cognitive aims - To be able to:
• build a sequence from a starting point to a
foreseen result
• compare the concept of angle as change in
direction with the Euclidean concept of angle as
space between two lines starting from a
common point
• compare the trace left by the turtle with the
Euclidean concept of edge
• compare the idea of circle as “the limit” of
regular polygons with the Euclidean definition of
circle as set of all points in the plane at a given
distance from a fixed point

13. Geometrical activities
The geometrical image of the world
 Renew the “ancient” Calculemus by Leibniz
and replace it by the new one Modelemus
(J.M. Laborde cit. in Cabri2004 – Rome,
September)
 Using Cabri II Plus for analysing real objects

14. Motivation…
Taking account of the pupils’ different attitudes and
learning styles with a variety of challenges and tools
( the rule of technology ….)
Beginning from pupils’ curiosities and interests
Or
Beginning from real world situations
Developing their own projects and sharing each
other ideas and knowledge (self guided learning,
cooperation)

15. We are working on:
•Looking around with a digital
camera
•Modelling with Cabri:
polygons, regular polygons,
geometric transformations
(symmetries, translations,
rotations…)
Examples
•Searching the geometry inside
real objects

16. What we expect
 The shift from a static knowledge to a
dynamic one
 A better integration between perceptive-
motor aspects and symbolic-representative
aspects
 A good training of geometric guessing

17. Geometrical constructions in triangles
(Orthocenter, Centroid, Incenter and Circumcircle)
Teacher F Tronchin
Dynamic Geometry versus by Hand Geometry
PUPILS INVOLVED
45 pupils in their 7th school year (2^ Media) in autumn
42 pupils in their 6th school year (1^ Media) in spring

18. BY HAND
Objectives:
1. To develop manual ability by using simple school tools
2. To improve precision in geometrical drawing
3. To reinforce the knowledge of some basic geometric
concepts like those of altitude of a triangle, angle
bisector, midpoint of a segment and perpendicular
bisector

19. BY USING THE COMPUTER WITH CABRI
(Dynamic geometry)
Objectives
1. To acquire the ability of using this software and its specific
terminology
2. To reinforce more dynamically the way to find the
orthocenter, centroid, incenter and circumcircle of a triangle
3. To develop the ability of evaluating the
correctness of the learned procedures

20. CONSIDERATIONS
 The two methods develop different abilities. We don’t want to
demonstrate which one is better or if we have to substitute the
traditional geometry with an interactive software. By using both we
want to develop complementary abilities and reinforce the knowledge
of some geometric concepts.
 In particular by using Cabri, because it gives the possibility to
build, correct and modify figures rapidly, pupils have a more relaxed
approach to geometry and their geometrical awareness probably
increases as well

21. CHEKS
1. A systematic noting of pupils’ impressions of and
reactions on the didactic activities
2. A final questionnaire to evaluate the pupils
experience and highlight positive and negative
aspects of the two learnng procedures
3. Possible final teachers’ remarks about the
effectiveness of the learned methodology

22. Geometrical constructions in triangles
(Orthocenter, Centroid, Incenter and Circumcircle)
By hand versus Cabri
At the end of the activity the pupils have been asked to answer 15
questions, among them the following:
Which ones are the best Cabri features?
 It is fun: 17
 It is fast: 15
 It allows me to move geometrical shapes and to understand how points do move:
16
 It makes me aware of what I am doing: 6
And the worse?
 It is automatic and does not make me think: 7
 It is boring: 1
 It is automatic and the next time I don’t remember how I have got there: 9
 There is nothing wrong with it: 5

23. Which ones are the best things about the manual work?
 The pleasure of building shapes a little at the time: 9
 The pleasure of working with pencil and manual tools: 6
 Because it is slow it makes me think how I am building it: 12
 None: 3
And the worse?
 It is too slow: 6
 It makes me insecure because if I make a mistake I need to start all over: 13
 The drawing is never perfect: 7
 No particular defects:3
Between the best Cabri features, the pupils talked the most about how fun, fast and
dynamic it is, while just few think that Cabri makes them more aware of what they are
doing. This agrees with the main defect they found in the software i.e. that it is too
mechanical.
Of the manual work it is appreciated most the fact that because it is slow it allows you
to think about what you are doing. But in the same time it is difficult to correct the
mistakes and it is imprecise.

24. TEACHER’S CONSIDERATIONS ABOUT CABRI
1. Without any doubts all the pupils have fun with it and this
reflects positively on the teaching and learning process
2. None of the pupils back off: i.e. the possibility this software gives to
correct easily and fast the mistakes make all the students to give it a try.
While in the manual work pupils who have difficulties may have hard time
and get stuck right from the beginning
3. The fact that the software is “dynamic” betters the awareness
especially among those that already works in a methodical way.
But pupils who find it difficult to draw, or work in a mechanical
way did not change much their way of working using the software

25. 4. It was a must for me to reduce the mechanical side of the software. In
order to do so, I constantly went over and over the basis of geometry, I
made the pupils reflect on why to choose one way instead of another. I
invited them to take measures of angles and sides to verify the validity
of their supposition.
5. The dynamic look of the software does not have any equivalent in any
hand made drawing and gives a big help in the learning process of the
basis of geometry which are better imprinted in the memory of the pupils
exactly thanks to the dynamic and visual look of it.

26. In CONCLUSION …
In agreement with the early statements and what the pupils say (almost all
think the methodologies are both good even if some of them would have
liked to work more with Cabri), it is my opinion that Cabri can not substitute
the traditional geometry. Anyway especially thanks to its dynamic feature it
represents an excellent tool to integrate the theoretical knowledge of
geometry and its practical aspects. And because it allows the students to
try several times it gives teacher the chances to have confrontations over
the mistakes they make.

27. Try and error
An introduction to “Equations”
 For 11-12 years old pupils

28. Try and error in a closed set
Additive pairs
x + y = N
N {0, 1, 2, 3….., N}
Which are the correct pairs ?

29. Try and error in a closed set
Multiplicative pairs
x * y = N
N {0, 1, 2, 3….., N}
Which are the correct pairs ?

30. Try and error
 An introduction to “Equations”
 For 13-14 years old pupils

31. Try and error: algebraic equations
Study an equation with a single variable
as:
ax+ p = bx +q
where (a,b,p,q) are real numbers

32. Try and error: algebraic equations
Search the solution of
the equation with
Cabri II plus

33. Science: hand–on activities
 We live in a “technological sea”: searching an
initial situation (from a problem in the daily
life)
Questions… Prior ideas
Selection of questions

34. Science: hand–on activities
Investigations
Direct experimentations conceived and realised by children
Material realisation
(search for a technical
solution)
Direct or instrument-aided
observation, with or without
measurement

35. Science: hand–on activities
Documentary research and divulgation
Pupils like teachers
Construction of know-how, knowledge and cultural
benchmarks

36. Science, hands–on activities
Investigations
Direct experimentations conceived and realised by children
Material realisation
(search for a technical
solution)
Direct or instrument-aided
observation, with or without
measurement
How a pile generates electricity

37. Science: hands–on activities
1. Which couples of
metals ….
Some questions during the hands-on activity:
2. Acid or basic or
neutral
environment…..
3. How to connect
the circuit….
….trying and checking different hypothesis….

38. Science: hands–on activities
At the end of activities……the best solution.

39. Science: hands–on activities
Aims
1. to stimulate and raise pupils' curiosity about the technological
world
2. to develop process skills, such as observing, questioning,
hypothesizing, predicting, investigating, interpreting, and
communicating, that play an essential role in helping children
develop a scientific way of thinking
3. to promote a direct, active participation of the pupils
4. to improve the children’s ability to work independently
5. to take account of the pupils’ different attitudes and learning
styles
6. to create a positive effect on the attitudes the pupils have for
science

40. Science: hand–on activities
From the mobile –
battery
To the lemon/tomato –
batteries
An example of activity

41. Science: hand–on activities and
mathematics
Slowbicycle race
Too slow!

42. This is an activity in which students (11-13 years old) compute the speed at which they rode
a bicycle.
Objectives:
In this activity students will:
 work in teams
 ride a bicycle as slowly as possible
 use a stopwatch to measure time
 record data
 use multiplication and addition to determine team’s total distance and time
 use division to calculate team’s average speed
 compare average speed with other teams to determine which team was the slowest
 create a bar graph to depict each team’s average speed
Questions to Ask:
1. What was the most difficult part about riding the bike?
2. What happens to the speed as the time increases?
3. What happens to the speed as the time decreases?
Travel Book Activities:
 Writing About a Trip
Slowbicycle race

43. Science activities
Some experiences on:
 Electrostatic
 Elementary electric
circuits
 Elementary
electromagnetism
With the use of everyday materials

44. Science activities
Practical executions with:
1. A representation, either
schematic or axonometric
2. A design with the proper
dimensions, writing a list
of the necessary
materials…..
At last the construction and the testing

45. Science activities
Using of ICT to communicate the results to
other pupils:
 Short videos
 hypertext

46. Dynamic Geometry versus by Hand Geometry
Teacher F. Tronchin
Science: Hands-on Activities in Physics
Teacher GP Moroldo

47. What factors
(mass, length, angle of
oscillation, shape…)
affect the period T
of oscillation of a
pendulum?
THE PENDULUM

48. To answer this question
it was important to...
1) …conduct tests always on
small oscillations;
2) …use heavy and dense objects in
order to make friction and
resistance negligible;
3) …vary one factor at a time.

49. Pendula with
different masses
but the same shape
and the same
length have allowed
us to exclude that
the mass is a
significant variable
in determining T.
The sample weights for scales were o.k.
... but lead fishing weights were our favorites!

50. We easily verified that length is the most important
variable: the change of T is evident!
…the
fastest→
…the
slowest
→
The shortest…
The longest…

51. MAIN DIFFICULTIES:
1. It’s hard to catch the
initial and final time of
an oscillation
2. When the length of the
pendulum is short, the
period is too brief to
observe!
SOLUTIONS:
 time several consecutive oscillations (usually 10)
 five or six people perform the same test in order to make an
average of the times
(possibly discarding “wild” data).

52. Andrea: 2.61
Alex: 2.60
Sara: 2.63
Luca: 2.59
Francesk: 2.64
Gucci: 2.66
G.Paolo: 2.81
(discarded)
The arithmetic mean
of acceptable values is
2.62 s and it will be used
in subsuquent data
processing…
For example, when L = 1,70 m we timed (s)

53. RESULTS
The results of our experiments
are shown in the table on the right.
It’s clear that when L increases, T also increases… but which
mathematical function can represent this growth?
For example, can we predict the period of a pendulum 3.5 m long?
(our ceilings are not high enough!)
L (m) T (s)
0.00 0.00
0.10 0.71
0.16 0.81
0.18 0.83
0.21 0.93
0.23 0.97
0.32 1.12
0.53 1.45
0.78 1.78
0.86 1.84
1.09 2.08
1.14 2.16
1.39 2.36
1.70 2.62
1.72 2.63
1.79 2.71
1.91 2.77
1.94 2.78
2.08 2.89
2.20 2.98
2.42 3.14
2.76 3.30
3.00 3.46
The pair (0, 0) was actually
added by us becouse on
decreasing of the length
the period T becomes
smaller and smaller…

54. 0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
3
3.5
L (m)
T (s)
The couples (L ;T) have been analyzed
by an appropriate software(*)
which
recognises the mathematical function
that represents growth.
(3.5 m ; 3.7 s)
(*) Graph 4.3 http://www.padowan.dk/graph/
We can forecast, for example, that
a pendulum 3.5 m long will have a
period T of about 3.7 s

55. L (m) T2
(s2
)
0.00 0.00
0.10 0.50
0.16 0.66
0.18 0.69
0.21 0.86
0.23 0.94
0.32 1.25
0.53 2.10
0.78 3.17
0.86 3.39
1.09 4.33
1.14 4.67
1.39 5.57
1.70 6.86
1.72 6.92
1.79 7.34
1.91 7.67
1.94 7.73
2.08 8.35
2.20 8.88
2.42 9.86
2.76 10.89
3.00 11.97
We obtained a line that describes how T changes
when L changes.
We are used to study mainly straigth lines, some
examples of hyperbola and of parabola… but this
is so different!
… we had a nice
surprise when we
tried to discover
how the length
determins the
squared period!

56. 0 1 2 3
0
1
2
3
4
5
6
7
8
9
10
11
12
L (m)
T2
We finally found something
familiar: we recognised that T2
is
directly proportional to L and the
constant of proportionality is 4
T2
= 4.
L
(0 ; 0)
(1 ; 4)
(2 ; 8)
(3 ; 12)
(0.5 ; 2)
(1.5 ; 6)

57. Our work concerned only a small
part of what pendulum studies
can suggest…

58. SOME IDEA ABOUT WORKING WITH PENDULUM…

59. Science: CO2 – an
environtmental problem
What can we study about CO2 in a
school lab, and how?
CO2 and living beings CO2 and rocks
CO2 and human activities
How these different
points are linked
between them?

60. Science:
CO2 – and living beings
Experiences on:
respirations fermentations
photosyntesis
Verifying the production or the
consume of CO2 in different
living beings:
•Human beings
•Yeasts
•Green plants
Beer yeasts, used for home pizza

61. Science:
CO2 – and rocks
Experiences on the
natural deposits of
CO2:
From the gas to the rock
From the rock to the gas

62. Science:
CO2 – and human activities
 The chemistry of combustion
 Global warming – greenhouse effects –
CO2……a ” carbon footprint calculator “
A website to analyse our CO2 emissions
http://www.safeclimate.net/calculator/index.php

63. The first input

64. The second input about
our home transportation

65. The answer by the CO2 calculator
for an input

66. 1. Is it possible to use the CO2
calculator in different classes?
2. What is the perception of the climate
changing in the families (interviewing
old people, collecting old photos…)
3. What are the data on the climate
changing in the different countries?
Science:
CO2 – and human activities

67. An hypothesis of cooperation between
classes:
using a Skype conference to describe
their researches to another class.
Science:
CO2 – and human activities